Hypoellipticity for a class of infinitely degenerate elliptic operators

Morimoto, Yoshinori; Morioka, Tatsushi

doi:10.1007/978-1-4612-2014-5_13

Cited by 10 publications

(15 citation statements)

References 26 publications

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“…* If λ(τ )/ log τ → ∞ as τ → +∞, then L satisfies a superlogarithmic estimate, hence is C ∞ hypoelliptic [4,13]. * If λ(τ )/ log τ → ∞ as τ → +∞, then L satisfies a superlogarithmic estimate, hence is C ∞ hypoelliptic [4,13].…”

Section: Remark 13mentioning

confidence: 99%

Hypoellipticity of the Kohn Laplacian for Three-Dimensional Tubular Cauchy–riemann Structures

Christ¹

2002

Journal of the Institute of Mathematics of Jussieu

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A necessary and sufficient geometric condition is established for the Kohn Laplacian to be hypoelliptic, modulo its nullspace, for boundaries of arbitrary infinitely differentiable pseudoconvex tube domains in two complex variables. Hypoellipticity is also proved to be equivalent to validity of a superlogarithmic estimate, for this special class of structures.By a tubular three-dimensional Cauchy-Riemann (CR) structure we will mean a CR structure defined in an open subset of R 3 , together with a coordinate system (x, y, t) ∈ R 3 , together with a CR operator of the formwhere φ ∈ C ∞ (R) is real-valued. Such CR structures may be realized as the boundaries of tube domains {z : Im z 2 > φ(Re z 1 )} in C 2 . The Levi form may be identified with the function φ (x). We always assume that φ is convex, so that the structure is pseudoconvex. By∂ * b we mean the adjoint of∂ b with respect to L 2 (R 3 , dx dy dt); thus∂The purpose of this note is to characterize hypoellipticity of the Kohn Laplacian∂ b∂ * b for this class of CR structures.Main Theorem. For any C ∞ pseudoconvex tubular three-dimensional CR structure, the following four conditions are equivalent in any open set.(α)∂ b is C ∞ hypoelliptic, modulo its nullspace.(β) The CR structure is not exponentially degenerate.(γ) The pair {∂ b ,∂ * b } satisfies a superlogarithmic estimate.

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Section: Remark 13mentioning

confidence: 99%

Hypoellipticity of the Kohn Laplacian for Three-Dimensional Tubular Cauchy–riemann Structures

Christ¹

2002

Journal of the Institute of Mathematics of Jussieu

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“…Related and more recent results include those of Kusuoko and Strook [KuStr85], Morimoto [Mori87], Christ [Christ95] and Bell and Mohammed [BellMo95]. Here, thanks in partt to helpful conversations with A. Bove, we will give a flexible and utterly elementary proof of Fediȋ's result which proves hypoellipticity in the smooth, Gevrey, and real analytic categories rapidly, when appropriate.…”

Section: Introductionmentioning

confidence: 82%

An elementary proof of Fediĭ’s theorem and extensions

Tartakoff

2006

Phase Space Analysis of Partial Differential Equations

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“…Since R S , m S independent of d, Proof of Proposition 5.1. For l b 1 fixed, we choose the functions of C y 0 ðWÞ as in [15,16],…”

Section: Nonlinear Hypoellipticitymentioning

confidence: 99%

“…The simplest example for (1.1) is the system in R 3 such as X 1 ¼ q x 1 , X 2 ¼ q x 2 , X 3 ¼ expðÀjx 1 j À1=s Þq x 3 with s > 0 (see [14,15,16]). The operator 4 X for this example degenerates infinitely on G 0 ¼ fx 1 ¼ 0g.…”

Section: Introductionmentioning

confidence: 99%

“…1) It should be noted that the controllability follows only from the hypoellipticity of 4 X þ cðxÞ. Conversely, it is known (see [14,15,16,18]) that the controllability does not imply the hypoellipticity of 4 X in general. The first example with 0 < s a 1 given at the beginning of this introduction, satisfies the controllability but not the hypoellipticity.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Hypoellipticity of Infinite Type

Morimoto

2007

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Abstract. We study the regularity of weak solutions for a class of second order semilinear infinitely degenerate elliptic equations. We get the regularity of weak solutions up to the boundary for Dirichlet problem, by noting the logarithmic regularity estimate for a linear principal part. In relation to this linear part, we also show the controllability and strong maximum principle for second order hypoelliptic operators even in the case where they degenerate infinitely. Model equations naturally come from some variational problems, if one replace the Laplace operator in such as Yamabe problems by degenerate elliptic operators. In the infinitely degenerate case, a permissible nonlinear term is not fractional power, compared with elliptic or subelliptic case. To treat this nonlinear term, the nonlinear microlocal analysis is developed in the logarithmic Sobolev space.

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Hypoellipticity for a class of infinitely degenerate elliptic operators

Cited by 10 publications

References 26 publications

Hypoellipticity of the Kohn Laplacian for Three-Dimensional Tubular Cauchy–riemann Structures

Hypoellipticity of the Kohn Laplacian for Three-Dimensional Tubular Cauchy–riemann Structures

An elementary proof of Fediĭ’s theorem and extensions

Nonlinear Hypoellipticity of Infinite Type

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